Double Dividend

© P. Berck 2008

Sources

• Goulder, Parry, Burtraw. Rand 1997• Fullerton. AER 1997• Fullerton and Metcalf. NBER wp 6199

1997

Pictures

L.0

Dirty good

X.0

tl

Private mc

mc

Labor Demand

Income tax distorts labor market while externality distorts goods market

GPB Model

• 3 Goods• Dirty X• Clean Y• Leisure H

• Dirty good externality • PPF: T=X+Y+H• Producer prices are all 1. • T – H is labor

• Taxes• tX for X

• tl for T-H, labor

• Gov’t revenue• TR= tl(T-H) + txX

• Given back to consumer lump sum.• Is constant

Consumer Problem

• Consumer problem max U(X,Y,H) • s.t. (1+tx)X + Y =(1-tl)(T-H) + TR

• good X costs more than good Y• labor (T-H) is taxed at rate tl

• foc: Ux=(1+tx) l; UY= l; UH=(1-tl) l l is marginal utility of income• Demands are X(tx,tl), Y(), H().

• Write X(t), Y(t), H(t) for short.

Consumer prices for Goods

• Approx 1/(1-tl) as 1+tl

• Budget constraint is then• (1+tl)(1+tx)X + (1+tl) Y =(T-H) + TR (1+tl)

• So is equivalent to a tax on both goods and a subsidy on TR

• (not appealing, but shows that Y really isn’t “untaxed”.

More setup

• Gov Rev Constraint + Budget imply PPF• just substitute for TR in budget

• Demands Equations satisfy Budget by construction

• So only one equation remains• TR= tl(T-H(t)) + txX(t)

• Taking the total derivative and rearranging give

Effect of tax increase on x

x ll x x

xl

x

dX Hx t t

dt dt tHdt T H tt

Social Problem

• U() + V(Q(X))• utility plus negative contribution from dirty

good.• V doesn’t enter into consumer choice

because it is aggregate X, not individual X that impairs breathing

Change in utility

• D = 1/ l V’ Qx

• Num of M is (1+t) – 1 times lost hours; partial equilib welfare loss

• Denom is partial equilib increase in tax rev from increase in labor tax

ll

ll

Htt

MH

T H tt

Intermediate Steps

'xx

dUU V Q

dt

'x x Y Hx x x x

dU dX dY dHU V Q U U

dt dt dt dt

Now substitute: l (1+tx) for Ux and so on.And D l for V’Qx (and note the sign reversal! My error, their error?And totally differentiate the ppf to get: dY/dtx = - dH/dtx- dX/dtx

Putting this together with the definition of M gives the final expression on The next slide

1

(1 )

xx x

xx

lx

dU dXD t

dt dt

dXM X t

dt

HM t

t

Comments

• Empirical applications are via CGE’s, which have lots of other things in them.

• When one raises a tax on labor it is equivalent to taxing both goods, to tx is the difference in the tax rate between the two goods with tl normalized to one.

• A standard doesn’t have the revenue recycling effect, cause there is no revenue.

• The pigouvian tax is probably not the right tax, though one can argue for too low or too high, depending on parameters. Goulder says too high.

The Dual

• DM notation.• PPF: p’y = 0

• (sign of work is negative, of goods positive)• Simple version has p fixed

• Budget: q’x = 0• gov’t budget: R= p’z = (q-p)’x

• Treat z as fixed

• 3 equations

Down to one eq.

• R = (q-p)’x• Let x = x(q-p) = x(t), the demand equation.• x(t) always satisfies x(t)’q = 0.

• R = t’x(t)

Feasible tax Variation

'

'j i

i

j

t xdt t

t xdtt

• W = V(q) – Dxi(q)• Indirect utility less damage• a is the marginal util of income• dV/dt = dV/dq = - a x by Roy’s identity

What happens when only taxes i and j are perturbed. Like tax on dirt up and labor down.

• Double dividend means first term is non-zero and original tax system is nonoptimal.

[ ] ( )j i i ii j

i i i j

dt x x dtV x x D

dt t t dt

• t = q-p• p is constant, so can write • q(t) = q(t+p) as the demand system• q(t) satisfies budget constraint by

construction

1 Equation left

Direct Approach

• Form the indirect utility function• IN(X(t),Y(t),H(t))= IN(t)

• Use Roy’s identity to get• dIN/dtx = -aX –aH dtl/dtx

• Adding the Pigou term• dU/dtx = -aX –aH dtl/dtx + V’Qx dX/dtx

• Here the dwl in X market decreases by aX dX; in labor market by –aH dtl/dtx dX.